$\large F(x) = \frac{x}{x + \frac{x}{x + \frac{x}{x + \frac{x }{._{._.}}} } }$

If the value of the first derivative of $F(x)$ at $x = 1$ can be expressed as $\frac{a\sqrt b - b}{c}$ for integers $a,b,$ and $c$, find the value of $a+b+c.$

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